[01] Mohammad Shafique, Department of Mathematics, Gomal University, Dera Ismail Khan, Pakistan.

The growth of a spherical amorphous crystal in supersaturated melt is analyzed numerically. The model problem is written in a scaled form which is suitable for numerical solution. Critical radii for the onset of instability to Yl, m bumps are obtained through simulation. These numerical results agree with those obtained previously for limiting values (very small and very large) of the super saturation. Moreover they show that the usual sequence of instabilities (Yl, m followed by Yl+1, m bumps) predicted by the classic Mullins and Sekerka model (valid only for small super saturation) breaks down for large super saturation.

Numerical Analysis, Spherical Crystal, Supersaturated Solution

[01] Chadam, J., Howison, S. D. and Ortoleva, P. (1987). IMA, Journal of Applied Math. 39, 1-15. [02] Mullins, W. W. and Sekerka, R. F. (1963). Journal of Applied Physics. 34, 323-329. [03] Mullins, W. W. and Sekerka, R. F. (1964). Journal of Applied Physics. 35, 444-451. [04] Krukowski, S. and Turski, L. A. (1982). Journal of Crystal Growth. 58, 631-762. [05] Chadam J. and Ortoleva, P. (1983). J. I. M. A. 30, 57-66. [06] Chadam, J., Billon, J. C., Bertsch, M., Ortoleva, P. and Peletie, L. A. (1985), College de France Seminar Vol. VI Research Notes in Mathematics 109. Pitman, London. [07] Chadam J. and Ortoleva, P. (1980). Proceedings of the Conference on Mathematical Biology, Carbondale, Illinois. [08] Rubenstein, L. I. (1971). The Stefan Problem. Translations of Mathematical Monographs 27, AMS, Providence. [09] Shafique, M. and Abbas, M. (1998) G. U. Journal of Research, 18, 89-94. [10] Shafique, M. (1989). M.Sc. Thesis. McMaster University, Hamilton, Canada. [11] Andrews, L. C. (1985). Special Functions for Engineers and Applied Mathematics, Macmillan Publishing Company, NY. [12] Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1986), Numerical Recipes. Cambridge University Press, Cambridge. [13] Jones, W. B. and Thorn, W. J. (1980). Continued Fractions: Analytic Theory and applications, Addison-Wesley, Reading. [14] Whittaker, E. T. and Watson, G, N. (1927). A course of modern analysis. University press, Cambridge. [15] Abramowitz, M. and Stegun, I. A. (1964). Hand book of mathematical functions. U. S. Government printing office, Washington, D.C. [16] Davis, P. J. and Rabinowitz, P. (1984). Methods of Numerical Integration. Academic Press, Inc. [17] Carslaw, H. S. and Jaeger, J. C. (1959). Conduction of Heat in Solds. Clarendon, Oxford [18] Friedman, A. (1964). Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs. [19] Friedman, A. (1982). Variational Principles and free Boundary Problems. Wiley, New York. [20] Elliott, C. M. & Ockkendon, J R. (1982). Weak and Variation methods for moving boundary problems. Pitman, London.